/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![25, 5, -11, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, w}>;

primesArray := [
[5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7],
[5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4],
[11, 11, w + 1],
[11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2],
[16, 2, 2],
[19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1],
[19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w],
[19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w],
[19, 19, w - 1],
[29, 29, w^2 - w - 8],
[29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2],
[31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2],
[31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3],
[61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5],
[61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15],
[61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3],
[61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13],
[71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4],
[71, 71, w^2 - 8],
[79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22],
[79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9],
[81, 3, -3],
[101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15],
[101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16],
[121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1],
[131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8],
[131, 131, w^2 - 2*w - 2],
[149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5],
[149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6],
[151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3],
[151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8],
[151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2],
[151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9],
[179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6],
[179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6],
[211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10],
[211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2],
[229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11],
[229, 229, 2*w^2 - 2*w - 9],
[229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2],
[229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3],
[239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19],
[239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3],
[241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6],
[241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1],
[251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12],
[251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15],
[269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2],
[269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9],
[271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w],
[271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2],
[281, 281, w^3 - 3*w^2 - 4*w + 13],
[281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6],
[281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16],
[281, 281, w^3 + w^2 - 8*w - 9],
[289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5],
[289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8],
[311, 311, -w^2 + 2*w + 11],
[311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1],
[311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1],
[311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10],
[349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16],
[349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7],
[359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1],
[359, 359, -w^3 + 3*w^2 + 5*w - 12],
[359, 359, w^3 - w^2 - 7*w + 2],
[359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11],
[379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9],
[379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7],
[389, 389, w^2 - 3*w - 8],
[389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16],
[401, 401, -3*w^2 + 2*w + 23],
[401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11],
[409, 409, -w^3 + 2*w^2 + 3*w - 7],
[409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2],
[419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2],
[419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w],
[421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3],
[421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3],
[439, 439, -w^3 + 3*w^2 + 5*w - 16],
[439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7],
[439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11],
[439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10],
[449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5],
[449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15],
[461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4],
[461, 461, 2*w^2 - 2*w - 11],
[461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4],
[479, 479, -w^3 + 3*w^2 + 6*w - 14],
[479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3],
[499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6],
[499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17],
[509, 509, -w^3 + 2*w^2 + 4*w - 6],
[509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4],
[521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7],
[521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8],
[529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15],
[529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21],
[541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10],
[541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13],
[541, 541, -2*w^2 + w + 13],
[541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2],
[569, 569, w^3 - 2*w^2 - 5*w + 7],
[569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6],
[571, 571, w^2 + w - 9],
[571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12],
[571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11],
[571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4],
[601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11],
[601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1],
[631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30],
[631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15],
[661, 661, -2*w^3 + 17*w + 12],
[661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1],
[691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35],
[691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19],
[701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6],
[701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8],
[709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18],
[709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20],
[709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5],
[709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10],
[719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12],
[719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1],
[719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14],
[719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8],
[739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5],
[739, 739, w - 6],
[761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11],
[761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w],
[769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8],
[769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15],
[769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18],
[769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5],
[809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14],
[809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4],
[809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5],
[809, 809, w^3 - w^2 - 5*w + 4],
[821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30],
[821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15],
[829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10],
[829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19],
[829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10],
[829, 829, -w^3 + 4*w^2 + 5*w - 23],
[839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4],
[839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8],
[841, 29, -w^3 + w^2 + 6*w - 4],
[859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9],
[859, 859, w^3 - 6*w - 3],
[881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1],
[881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14],
[881, 881, -w^3 + 6*w + 2],
[881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6],
[911, 911, w^3 - w^2 - 5*w + 1],
[911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2],
[919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7],
[919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5],
[929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16],
[929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7],
[961, 31, w^3 - w^2 - 6*w + 2],
[971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6],
[971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w],
[971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7],
[971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10],
[991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7],
[991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 34*x^4 + 26*x^2 - 5;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, -1, e, 3*e^5 - 101*e^3 + 44*e, -5*e^5 + 168*e^3 - 64*e, -8*e^5 + 269*e^3 - 107*e, 2*e^4 - 67*e^2 + 20, -5*e^4 + 168*e^2 - 65, 5*e^5 - 168*e^3 + 64*e, 3*e^4 - 101*e^2 + 40, 5*e^5 - 168*e^3 + 61*e, -2*e^5 + 67*e^3 - 18*e, -e^5 + 34*e^3 - 25*e, 6*e^5 - 202*e^3 + 90*e, 4*e^5 - 134*e^3 + 36*e, -8*e^4 + 269*e^2 - 107, 12*e^4 - 403*e^2 + 153, -2, 3*e^4 - 101*e^2 + 33, -3*e^4 + 100*e^2 - 25, e^5 - 34*e^3 + 24*e, 9*e^5 - 302*e^3 + 99*e, -18*e^5 + 605*e^3 - 233*e, -4*e^5 + 135*e^3 - 71*e, -6*e^4 + 201*e^2 - 73, 31*e^5 - 1042*e^3 + 399*e, -15*e^5 + 504*e^3 - 189*e, 19*e^4 - 638*e^2 + 245, 21*e^5 - 706*e^3 + 274*e, 4*e^4 - 134*e^2 + 43, -13*e^4 + 437*e^2 - 172, -11*e^4 + 369*e^2 - 138, 12*e^4 - 404*e^2 + 157, 21*e^5 - 706*e^3 + 277*e, 16*e^4 - 538*e^2 + 205, -7*e^4 + 236*e^2 - 108, -14*e^4 + 471*e^2 - 183, -8*e^4 + 268*e^2 - 100, 3*e^4 - 101*e^2 + 20, -29*e^5 + 975*e^3 - 381*e, -19*e^5 + 638*e^3 - 229*e, 9*e^5 - 303*e^3 + 130*e, -12*e^4 + 403*e^2 - 155, 21*e^5 - 706*e^3 + 274*e, -37*e^5 + 1244*e^3 - 490*e, -14*e^5 + 471*e^3 - 197*e, -22*e^5 + 739*e^3 - 271*e, -10*e^5 + 336*e^3 - 123*e, -2*e^4 + 67*e^2 - 25, 7*e^4 - 235*e^2 + 68, -e^4 + 33*e^2 - 12, 4*e^4 - 134*e^2 + 42, 15*e^5 - 504*e^3 + 189*e, -25*e^5 + 840*e^3 - 311*e, -18, 26*e^5 - 874*e^3 + 340*e, 5*e^4 - 167*e^2 + 55, -11*e^5 + 370*e^3 - 155*e, -13*e^5 + 436*e^3 - 136*e, 22*e^5 - 739*e^3 + 272*e, 24*e^5 - 807*e^3 + 325*e, -5*e^5 + 169*e^3 - 94*e, -17*e^4 + 572*e^2 - 230, 16*e^5 - 539*e^3 + 244*e, -e^5 + 33*e^3 + 8*e, -2*e^4 + 68*e^2 - 35, -24*e^4 + 807*e^2 - 315, 26*e^4 - 874*e^2 + 335, -38*e^5 + 1277*e^3 - 486*e, -17*e^4 + 572*e^2 - 225, 37*e^5 - 1244*e^3 + 496*e, 32*e^5 - 1075*e^3 + 396*e, -37*e^5 + 1244*e^3 - 493*e, 6*e^4 - 201*e^2 + 70, -20*e^5 + 671*e^3 - 217*e, -47*e^5 + 1580*e^3 - 615*e, -17*e^4 + 571*e^2 - 240, 33*e^5 - 1109*e^3 + 418*e, 17*e^5 - 571*e^3 + 208*e, 10*e^5 - 337*e^3 + 163*e, -21*e^4 + 706*e^2 - 265, -35*e^4 + 1177*e^2 - 450, 18*e^5 - 605*e^3 + 232*e, 40*e^4 - 1344*e^2 + 515, -7*e^5 + 236*e^3 - 109*e, e^4 - 33*e^2 - 23, -16*e^4 + 538*e^2 - 232, 24*e^4 - 808*e^2 + 322, 23*e^4 - 773*e^2 + 300, -9*e^5 + 304*e^3 - 159*e, 7*e^5 - 234*e^3 + 55*e, 37*e^4 - 1243*e^2 + 470, -6*e^5 + 203*e^3 - 125*e, 5*e^4 - 168*e^2 + 85, -13*e^4 + 437*e^2 - 172, 27*e^4 - 909*e^2 + 358, 25*e^4 - 842*e^2 + 345, -57*e^5 + 1916*e^3 - 738*e, 30*e^5 - 1009*e^3 + 415*e, 15*e^5 - 505*e^3 + 221*e, -2*e^5 + 69*e^3 - 89*e, -4*e^5 + 134*e^3 - 32*e, -27*e^4 + 908*e^2 - 365, -16*e^5 + 537*e^3 - 179*e, 2*e^5 - 66*e^3 - 13*e, -8*e^4 + 270*e^2 - 127, -23*e^4 + 773*e^2 - 312, 21*e^5 - 705*e^3 + 244*e, 16*e^5 - 539*e^3 + 254*e, -70*e^5 + 2353*e^3 - 910*e, -6*e^4 + 201*e^2 - 83, -2*e^4 + 65*e^2 + 17, 49*e^5 - 1647*e^3 + 633*e, 15*e^5 - 503*e^3 + 157*e, 15*e^4 - 505*e^2 + 198, 13*e^4 - 437*e^2 + 188, 25*e^4 - 841*e^2 + 303, 15*e^4 - 503*e^2 + 163, 29*e^5 - 973*e^3 + 324*e, -24*e^5 + 807*e^3 - 325*e, -23*e^4 + 772*e^2 - 280, 7*e^4 - 234*e^2 + 75, -8*e^4 + 269*e^2 - 90, -17*e^5 + 570*e^3 - 168*e, -13*e^4 + 439*e^2 - 185, -e^5 + 33*e^3 + 15*e, -48*e^5 + 1614*e^3 - 645*e, -13*e^4 + 438*e^2 - 170, -73*e^5 + 2453*e^3 - 930*e, -37*e^5 + 1243*e^3 - 452*e, 24*e^4 - 806*e^2 + 300, e^5 - 35*e^3 + 65*e, 16*e^4 - 540*e^2 + 235, -23*e^5 + 772*e^3 - 267*e, -12*e^4 + 404*e^2 - 140, -25*e^5 + 839*e^3 - 285*e, 23*e^4 - 774*e^2 + 325, 44*e^5 - 1479*e^3 + 581*e, 3*e^5 - 101*e^3 + 43*e, 22*e^5 - 738*e^3 + 234*e, -27*e^4 + 907*e^2 - 340, -9*e^5 + 302*e^3 - 91*e, 16*e^4 - 536*e^2 + 185, -60*e^5 + 2017*e^3 - 784*e, -19*e^5 + 640*e^3 - 293*e, -11*e^4 + 369*e^2 - 130, 33*e^5 - 1109*e^3 + 424*e, -7*e^5 + 236*e^3 - 121*e, 16*e^4 - 538*e^2 + 195, -11*e^4 + 368*e^2 - 108, -41*e^4 + 1380*e^2 - 548, e^2 - 37, 8*e^4 - 267*e^2 + 63, 54*e^5 - 1815*e^3 + 706*e, 61*e^5 - 2050*e^3 + 783*e, 2*e^5 - 70*e^3 + 114*e, 2*e^4 - 66*e^2 + 30, -40*e^4 + 1343*e^2 - 500, 52*e^5 - 1747*e^3 + 651*e, 19*e^4 - 639*e^2 + 242, -34*e^4 + 1142*e^2 - 438, -26*e^4 + 874*e^2 - 357, 21*e^4 - 707*e^2 + 257, 5*e^4 - 169*e^2 + 98, 18*e^5 - 606*e^3 + 261*e, -e^5 + 35*e^3 - 64*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7}>] := -1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;